Nebraska officials said they're concerned about an apparent rush by parents to drop their teenage children off at hospitals before lawmakers change the state's troubled "safe haven" law.

"Please don't bring your teenager to Nebraska," Gov. Dave Heineman told CNN. "Think of what you are saying. You are saying you no longer support them. You no longer love them."

Nebraska's safe haven law was intended to allow parents to hand over an infant anonymously to a hospital without being prosecuted. Of the 34 children who have been dropped off at hospitals, officials said not one has been an infant. All but six have been older than 10, according to a Nebraska Department of Health and Human Services analysis.